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On the Asymptotic Expansion of the Kashaev Invariant of the 52 Knot

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On the Asymptotic Expansion of the Kashaev Invariant of the 52 Knot On the asymptotic expansion of the Kashaev invariant of the 52 knot Tomotada Ohtsuki Abstract We give a presentation of the asymptotic expansion of the Kashaev invariant of the 52 knot. As the volume conjecture states, the leading term of the expansion presents the hyperbolic volume and the Chern-Simons invariant of the complement of the 52 knot. Further, we obtain a method to compute the full Poincare asymptotics to all orders of the Kashaev invariant of the 52 knot. 1 Introduction h i 2 C ··· In [16, 17] Kashaev defined the Kashaev invariant L N of a link L for N = 2; 3; by using the quantum dilogarithm. In [18] he conjectured that, for any hyperbolic link L, log h L i 2π · lim N = vol(S3 − L); N!1 N where \vol" denotes the hyperbolic volume, and gave evidence for the conjecture for the figure-eight knot, the 52 knot and the 61 knot, by formal calculations. In 1999, H. h i Murakami and J. Murakami [24] proved that the Kashaevp invariant L N of anyp link L 2π −1=N 2π −1=N is equal to the N-colored Jones polynomial JN (L; e ) of L evaluated at e , where JN (L; q) denotes the invariant obtained as the quantum invariant of links associated with the N-dimensional irreducible representation of the quantum group Uq(sl2). Further, as an extension of Kashaev's conjecture, they conjectured that, for any knot K, p log jJ (K; e2π −1=N )j 2π · lim N = vol(S3 − K); N!1 N where \vol" in this formula denotes the simplicial volume (normalized by multiplying by the hyperbolic volume of the regular ideal tetrahedron). This is called the volume conjecture. As a complexification of the volume conjecture, it is conjectured in [25] that, for a hyperbolic link L, p p log J (L; e2π −1=N ) p 2π −1 · lim N = cs(S3 − L) + −1 vol(S3 − L) (1) N!1 N for an appropriate choice of a branch of the logarithm, where \cs" denotes the Chern- Simons invariant. Furthermore, it is conjectured [11] (see also [3, 12, 46]) from the view-p 2π −1=k point of the SL(2; C) Chern-Simons theory that the asymptotic expansion of JN (K; e ) 1 of a hyperbolic knot K as N; k ! 1 fixing u = N=k is given by the following form, p p ( X1 ( − ) ) 2π −1=k N& 3=2 2π 1 i JN (K; e ) ∼ e N ! · 1 + κi · (2) N;k!1 N u=N=k: fixed i=1 for some scalars &; !; κi depending on K and u, though they do not discuss the Jones polynomial in the Chern-Simons theory in the case of vanishing quantum dimension, which is discussed in [38]. These conjectures look interesting in the sense that they make a bridge between quantum topology and hyperbolic geometry, and it suggests the existence of a future theory between them. An approach toward a proof of the volume conjecture has been known, which is due to Kashaev [18], Thurston [35] and Yokota [42]. We briefly review this approach, as follows. h i By definition, the Kashaev invariant K N (which we review in Section 2.1) of a knot K is given byp a sum of fractions whose denominators are product of copies of (q)n, where 2 n q = exp(2π −1=N) and (x)n = (1 − x)(1 − x ) ··· (1 − x ); for example, the Kashaev h i invariant 52 N of the 52 knot is given by X 3 h i N q 52 N = ; (q) (q) − − − (q) (q) (q) 0 ≤ i;j i+j N i j 1 j j i i+j < N ( ( p )) ∼ pN − 2πn −1=N as shown in (7). By the approximation (q)n exp 2π −1 Li2(1) Li2(e ) , we expect the following approximation, X ( p p ) 3 N ˇ 2π −1 i=N 2π −1 j=N h 52 i ∼ N q exp p V (e ; e ) ; N ? 2π −1 0 ≤ i;j i+j < N where we put ( 1 ) (1) ( 1 ) Vˇ (x; y) = Li (xy) + Li + Li (y) − Li − Li − Li (1): 2 2 xy 2 2 y 2 x 2 Further, by formally replacing the sum with an integral putting t = i=N and s = j=N, we expect that Z ( p p ) 5 N ˇ 2π −1 t 2π −1 s h 52 i ∼ N q exp p V (e ; e ) dt ds: N ?? 0 ≤ t;s 2π −1 t+s ≤ 1 Furthermore, by applying the saddle point method, we expect that the asymptotic behav- ior might be described by a critical value of Vˇ . Yokota [42] showed that a critical value of such a function Vˇ is given by the hyperbolic volume of the knot complement. There are problems justifying this series of arguments. The volume conjecture has been rigorously proved for some particular knots and links such as torus knots [19] (see also [5]1), the figure-eight knot (by Ekholm, see also [1]2), 1A detailed asymptotic expansion of the colored Jones polynomial for torus knots is given in [5]. 2A detailed proof of the volume conjecture for the figure-eight knot was given in [1] and the term N 3=2 in (2) was also verified there. 2 Whitehead doubles of (2; p)-torus knots [47], positive iterated torus knots [37], the 52 knot [20], and some links [9, 15, 36, 37, 41, 47]; for details see e.g. [22]. They (except for the 52 knot) have particular properties; for example, the simplicial volumes of the complements of torus knots are 0, and a critical point of Vˇ for the figure-eight knot is on the original contour of the integral. The volume conjecture for them has been proved by using such particular properties. In particular, the 52 knot is of a general case; the volume conjecture for the 52 knot has been proved by Kashaev and Yokota [20] by presenting the above mentioned sum by the residue of a certain integral. The aim of this paper is to give a presentation of the asymptotic expansion of the h i Kashaev invariant 52 N of the 52 knot rigorously (Theorem 1.1 below). Let y0 be the unique solution with positive imaginary part of (y − 1)3 = y, p y0 = 0:3376410213::: + −1 · 0:5622795120::: : We put 1 x = 1 − ; 0 y 0 ( ) 1 ( 1 ) ( 1 ) ( 1 ) π2 & = p Li (x y ) + Li + Li (y ) − Li − Li − − 2 0 0 2 x y 2 0 2 y 2 x 6 2π 1 p 0 0 0 0 = 0:4501096100::: + −1 · 0:4813049796::: ; p ( ) − y0 − 1 1=2 p ! = eπ 1=4 = 0:0901905774::: − −1 · 0:6499757866::: : 2y0 + 1 Then, we have h i Theorem 1.1. The asymptotic expansion of the Kashaev invariant 52 N of the 52 knot is given by the following form, p ( d ) X (2π −1) ( 1 ) h 5 i = eN& N 3=2 ! · 1 + κ · i + O ; 2 N i N N d+1 i=1 for any d, where κi is some constant given by a polynomial in y0 with rational coefficients; in particular, κ1 is given by ( ) 1 37 31 35 1 κ = y2 − y + + 1 = (1650y2 − 3498y + 2197) + 1 1 (2y + 1)3 4 0 6 0 8 12696 0 0 0 p = 1:0537470859::: − −1 · 0:1055728779::: : −1h i −N& −3=2 We can numerically observe that the limit of q 52 N e N tends to the above mentioned value of !. −1h i −N& −3=2 N q 52 N e N p 200 0:0915851738::: − −1 · 0:6519891312::: p 500 0:0907489101::: − −1 · 0:6507787459::: p 1000 0:0904698237::: − −1 · 0:6503768725::: 3 p h i N& 3=2 −1− − Further, we can numerically observe that the limit of ( 52 N (e N !) 1)N=(2π 1) tends to the above mentioned value of κ1. p h i N& 3=2 −1 − − N ( 52 N (e N !) 1)N=(2π 1) p 200 1:0567234007::: − −1 · 0:0885918466::: p 500 1:0549811019::: − −1 · 0:0987904427::: p 1000 1:0543713307::: − −1 · 0:1021833710::: As the conjecture (2) suggests, ! and κi's of (2) are expected to be invariants of K for any hyperbolic knotpK. We have computed ! and κ1 for the 52 knot in this paper. It is conjectured that 2 −1 !2 of a hyperbolic knot is equal to the twisted Reidemeister torsion associated with the action on sl2 of the holonomy representation of the hyperbolic structure; see Remark 1.4 below. We discuss about it for some knots in [28]. We give a proof of the theorem in Section 5 by justifying the above mentioned approach. An outline of the proof is as follows. We rewrite the sum (7) by an integral by the Poisson summation formula. When we apply the Poisson summation formula, the right-hand side of the Poisson summation formula consists of infinitely many summands, and we show that we can ignore them all except for the one at 0 in the sense that they are of sufficiently small order at N ! 1 (Proposition 4.6 and Lemma 5.8). Further, by the saddle point method (Proposition 3.5), we calculate the asymptotic expansion of the integral, and obtain the presentation of the theorem. By the method of this paper, the asymptotic behavior of the Kashaev invariant is discussed for the hyperbolic knots with up to 7 crossings in [27, 26] and for some hyperbolic knots with 8 crossings in [34]. Remark 1.2.
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